Question \(W^{thick}_{{\mathbb {H}}^3}\)  appears to be related to the Geometric Koebe Conjecture, recently introduced by Luo and Wu [1]}. We first recall this conjectures, as well as recent and less recent results concerning them. Recall that a circle domain in \({\mathbb {C}P}^1\)  is the complement of a family of open round disks and points.
